Harbingers of Artin's Reciprocity Law. II. Irreducibility of Cyclotomic Polynomials

TL;DR

This paper discusses the connection between Artin's reciprocity law and the irreducibility of cyclotomic polynomials, highlighting a key step in the proof involving Dedekind's method.

Contribution

It establishes a link between the proof of Artin's reciprocity law and Dedekind's proof of cyclotomic polynomial irreducibility, clarifying a fundamental step.

Findings

Identifies the verification of reciprocity law with Dedekind's irreducibility proof

Clarifies the role of cyclotomic polynomials in class field theory

Provides historical context for auxiliary primes

Abstract

In the first article of this series we have presented the history of auxiliary primes from Legendre's proof of the quadratic reciprocity law up to Artin's reciprocity law. We have also seen that the proof of Artin's reciprocity law consists of several steps, the first of which is the verification of the reciprocity law for cyclotomic extensions. In this article we will show that this step can be identified with one of Dedekind's proofs of the irreducibility of the cyclotomic polynomial.